The future of maths

Maths, it’s everywhere! Whether it’s the computer in our cars or the smartphone we use everyday, maths plays an increasingly important role in our daily lives.

But what impact are digital technologies themselves having on the way the process of maths works? And in a digital age do we need to change the way we teach maths to make it more relevant to our lives in the 21st century?

Transcript

One of the things we like doing on this program, as I'm sure you're aware, is we like to occasionally take issues that are normally difficult to explore on radio and give them the Future Tense treatment.

In the recent past we've done algorithms, data, the future of academic journals, those kinds of enormously important, but not very sexy issues, from a conventional media point of view.

Today, we're doing it again. In this show we'll explore the future of maths, or 'math' as our podcasters in North America would say.

And the idea for this particular show came from listener Gordon Clarke.

Hello Gordon.

Gordon Clarke: Hello Antony, how are you?

Antony Funnell: Okay, in 30 words or less, why maths?

Gordon Clarke: Why maths? I think, Antony, there are two issues. We are on the verge of some big changes I think in maths. One, maths has been very much community-based over many, many centuries, but now it's happening even more so with the use of internet. We're getting people who are separated half a world apart being able to collaborate online in real time, throwing ideas backwards and forwards and completing proofs and things like that. So that is something that people had to either do a sabbatical or do via email in the past. Now the rules are all changing, things are happening at a much, much faster pace.

Antony Funnell: And point two?

Gordon Clarke: And point two is computers. We've got to the point now where computers are so powerful, you've got neural networks and other things like that running. When are we going to allow a computer to come up with a formal proof? We've actually had a paper published, I think in 1998, so it's only 14 years ago, of a mathematician who wrote a program and produced a proof for an old conjecture of Kepler's from back in 1611. How much more is that going to happen? I think that's going to happen a lot more, we're going to see things like computers giving mathematicians intuition or perhaps in discovering new relationships, or maybe even falsifying conjectures, things like that. I think the world is opening up for mathematicians in a huge way now.

Antony Funnell: Great, thanks for that Gordon. The future of maths is the theme for today's show, and how we should teach it. And I'm pleased to say we've got some big names in mathematics, including Conrad Wolfram and Fields medallist Terence Tao, so let's get going.

Conrad Wolfram: Maths is often implicit, it's inside lots of things that happen, it's not necessarily explicitly calculating, and particularly that the person on the end isn't hand-calculating. So we see maths, whether it's optimising our car when we're driving along or whether it's figuring out when our plane's going to schedule, whichever aspect, life has got far more quantitative and far more optimised because we're using high-powered maths or other people are using high-powered maths to run it.

And in a sense what's happened is that computers have liberated maths from hand-calculating. So what they've done is to take a subject that was kind of bound by what individuals could do on paper or in our heads, and it has given it the most amazing tool that any ancient subject has ever been given in the form of computers. And that has meant that we now apply maths in all sorts of places where we would never have dreamt of applying it 50 years ago.

Antony Funnell: That's Conrad Wolfram, the founder and managing director of Wolfram Research Europe which develops computer software for mathematics, and he's been on a bit of a crusade in recent years to get us to change the way we think about maths and the way we teach it. Too much classroom time, he argues, is still being spent teaching kids how to calculate, when computers can do it for them.

Now, if you're not involved in the field of mathematics, or maths teaching, that probably doesn't sound controversial, but you'd be surprised just how much argument Wolfram's position continues to stir up.

Conrad Wolfram: People confuse, if you like, the step of hand-calculating with the bigger subject of maths. You see, what I think maths is is asking questions about the world, asking what is it that makes such-and-such risky. House prices in Australia and the UK, are they connected or are they not connected? How do we model that? And then the next step is to move that onto putting it in maths form, putting it in this powerful language of mathematics that allows us to work out results and problem-solve. And that's what I call step two.

And then the step three is this hand-calculating or computer calculating, it's calculating in one form, taking it from the way you set it up, to some answer in a mathematical form. And then step four is going back the other way and verifying that answer, making sure it makes sense, turning it back into solving the problem.

And the real thing that people see in schools is step three, having to do endless calculation by hand, yet that's probably the steps that we should least be seeing as humans because we can get computers to do it so much better than us. But on the other hand, we should be doing much more of the problem set-up, putting it into maths and taking it back out of a maths and verifying it, which humans are still for the most part better at doing.

Antony Funnell: It's a curious thing then, isn't it, that while we've had an explosion in the usage of computers in recent decades, that they haven't really come into the mass classroom in the way that you would like them to. And there is almost a sense that to use computers, to solve some of these captivating problems is actually almost cheating, it's almost robbing ourselves of a skill.

Conrad Wolfram: Yes, I think there are a couple of different confusions that generate this impression. The first one is that somehow computers should replace teachers as the primary thing rather than fundamentally change the subject. Now, I'm all for modern techniques of using computers to assist teaching, to assist the process of learning, and that's fine, but in maths we have a specifically different issue which is the subject matter of mathematics itself changed, and it changed because computers fundamentally reformed the subject matter in the outside world, and yet we haven't reflected that in education because we still have this notion of mathematics as being all about hand-calculating. So that's one confusion level.

And so then what happens is people then assume that you need...that somehow a computer is going to solve your problems, when in fact the big problem to solve is how to set up those mathematical problems and get a reasonable answer out of them, how to deal with the results. So that's one area that I think is confusing.

Another thing is to do with maths itself. Computer science and maths diverged in the real world. There were a set of mathematicians who didn't really want computer science to be part of maths, at the very early stages back in the '50s, and I think that divergence hasn't helped in education. And one of the issues we have right now is the users of mathematics are perhaps 95% not classified as mathematicians, they are scientists or engineers or artists or architects, many different areas. So they are not really mathematicians in their own right, and yet we have mathematicians setting most of the curricula. So they have a rather narrow view in many cases of what people actually need in this compulsory subject that we push everyone into doing.

Antony Funnell: The criticism that I've seen of your idea is that it will lead to dumbing down, that you'll end up with students who don't really understand the very basics, that they need the basics in order to advance on. What would you say to that?

Conrad Wolfram: I'd say first that the problems we're setting students right now is dumbed down, so the base we're starting from is a low base. And if you look at the problems, the problems are simplistic and they are simplistic because everyone has to be able to hand-calculate them, but yet that's not what happens in the real world. So we have people jumping in a particular way, but you wouldn't ever really calculate that, you would calculate something much more complicated. The principles that we're trying to get students to apply are in fact the same in the more complicated cases and the simpler cases, but they seem very unreal and artificial. And the actual complexity of the problem, the conceptual interest in many cases, is in putting that complex problem in a mathematical form.

The idea that the real world has become conceptually simpler or dumbed down because computers somehow make it that way in science and engineering and biology and all these other areas where we use mathematics, the idea that they've become conceptually simpler since computers did the calculating is nuts, in my view, and there's no reason why that should be the case in education, far from it. And I think we're between two schools at the moment because people say that if you do long division by hand, that somehow that's conceptually empowering. I don't really understand why that is; you're applying a process that you don't really understand in most cases, you're rote learning a process, and that doesn't seem particularly conceptually empowering. Answering complex questions about the world outside using mathematics as a tool seems far more conceptually empowering and in a sense far more intellectual.

Antony Funnell: I have to ask you about a second criticism that I've seen, and that is that we would expect somebody to say what you're saying given that the organisation you work for is involved in the development of maths software, that there is a conflict of interest. How would you answer that?

Conrad Wolfram: Potentially there is, but so there is for everyone in this debate. You know, if you're a mathematician who has learned traditional maths of the last 30 years, you have a vested interest too. So what I ask people to do is listen to the power of the arguments one way or another. I would also say that trying to make money out of high schools as opposed to many of the other places that we sell our Mathematica software, for example, and whether that is in industry or government, which is where much of our business comes from, those are far easier markets to tackle, I can assure you, in terms of making money, if that were the objective, than schools.

Antony Funnell: Conrad Wolfram from Wolfram Research Europe.

Keith Devlin: It is not clear that mathematics will change quite as fast as technology, but it's certainly changing faster, and when it makes these changes not only do those of us in the business have trouble keeping up, but people outside have trouble keeping up, and the educational world is going to have a real problem, because if you look at the way we teach mathematics in our school systems, almost everything that we teach our kids up to the age of about 14 or 15 was set in place by the 9th and 10th centuries. The rest of it, the higher level stuff we teach our kids when they are about to leave high school, namely calculus, that is 17th century. There is virtually nothing from the 18th century onwards which we teach our kids. Well, you think how much the world has changed in 20 years, let alone 300 years, then there is hardly any wonder that most people today have a great deal of difficulty with the kind of mathematics that is required because they never, ever see it in the school system.

Antony Funnell: That's Dr Keith Devlin, a science communicator for NPR in the United States and the director of Stanford University's Human-Sciences and Technologies Advanced Research Institute.

Like Conrad Wolfram, he believes the way many of us think about maths and teach it no longer matches the reality of the 21st century digital world.

Keith Devlin: We're used to the fact that technological developments, hardware building, software developments, much of that is outsourced, and that involves a lot of mathematics. So when you outsource the building of an iPad or something, you are in a sense outsourcing some of the mathematics. What is more intriguing to me is that the actual mathematics itself is getting outsourced, not just the mathematics in hardware design and in the software writing, but explicit mathematics, solving equations, formulating equations, developing spreadsheets and so forth, drawing graphs, that kind of thing is increasingly getting outsourced as well. And that's the kind of thing that people I think are not yet aware of. We still think that it's important that people learn how to do...in Western countries and in countries like Australia, we still think it's important to do mathematics, and in a way it is, but the mathematics that we need them to learn to do isn't what it was when I was a young kid and when lots of other people were younger.

The stuff that I learnt at university and that was very useful for me in building a career, most of that can actually be outsourced and done much more cheaply elsewhere. So it's not that mathematics is going to go away, it's not that the future won't require mathematics, but the mathematics that people need to learn to do well in the advanced nations of the world, that has changed.

Antony Funnell: So what should we be teaching our children? What sort of maths should we be teaching our children in countries like the United States and Australia?

Keith Devlin: It's the high-level mathematics. For example, when I was a kid, it was actually very important to learn how to do arithmetic because I grew up at a time when there weren't even calculators. The best thing we had was a slide rule. So if you weren't good at arithmetic, then you weren't going to be successful. But once we got pocket calculators, that went away, and so we're used to the fact, those of us who are more than 30 or 40 years old and I'm a bit older than that, are used to the fact that things that used to be important no longer are important.

The knowledge and the ability to do arithmetic fast and accurately in your head went away. That didn't mean to say you weren't good at numbers. You had to be sophisticated about numbers but you didn't have to sit down and do that long division because the calculator did it much more quickly and more efficiently and more accurately than a person could. So you have to learn to be sophisticated users of the technologies. You can't make a good use of the calculator unless you have very good number sense.

But the emphasis shifts. And incidentally, this is why, at least I know in the United States and I suspect in Australia as well, there has been a great emphasis on being good at algebra. That ends up getting mangled into being able to solve quadratic equations, which isn't doing algebra at all. Solving quadratic occasions is actually arithmetic. What you have to be good at today is reasoning not with numbers, not doing calculations with numbers but reasoning about numbers.

For example, most of us in our professions or our daily lives or in our recreation find we actually need to do things with spreadsheets, if it's keeping track of our finances or the scores in our sports club or whatever it is, or our businesses, most of us find we use spreadsheets a lot. If you use a spreadsheet you don't have to do the arithmetic, the computer does that. What you have to do is build the spreadsheet. That's algebraic thinking. The reason societies are now saying algebra is important is because creating and using a spreadsheet is algebra. And the shift has gone from the arithmetic, which is now automated, to using that spreadsheet that does the arithmetic for you. We've ratcheted up one level the kind of thinking we have to do. When you use a spreadsheet you're not doing calculations with numbers, you're thinking about patterns of numbers, and that makes it algebra.

Antony Funnell: Our next guest today in our look at the future of mathematics and maths teaching is noted mathematician Jon Bowein. Borwein is laureate professor of mathematics at the University of Newcastle in New South Wales and he's also a former president of the Canadian Mathematical Society, among other things.

Now, my question to Professor Borwein is this; just how useful are computers for advancing mathematics itself? And on that score, Professor Borwein says the jury is still out.

Jonathan Borwein: It gets said in a profession that we invented the computer 60, 70 years ago and then ignored it for 50 years. So it's only in the last 20 to 25 years that computers have been large enough, have had sophisticated enough software that can really do something that looks like mathematics in the way that…I think you've talked to Conrad Wolfram…that Wolfram software or maybe Maple do.

So we are able to look at much bigger problems, we're able to look at them in different ways. It's very exciting to do things that are data mining or simulation, but there is also something in mathematics called the cursive exponentiality, which is that you can ask a question and you can do the first case...say it has 'when N=2', it's a simple question, you can do it maybe in your head, and when N=3 you can do it with a bit of work on paper, N=4 you can do it in an hour on a computer, N=5 the supercomputer struggles and gets an answer, and N=6 you'd need the rest of the universe in computer time.

And maybe you'll see the entire pattern from the first four cases, in which case that's great, if not all you've done is bump into the fact that many, many mathematical problems are much too hard to put on computers.

At the other end (and this is where I'm more interested) is what I call micro parallelism; the ability to ask a lot of simple questions and get quick answers, which is really what something like Google does when it has transformed the way we ask for information and what we expect to be able to see.

Antony Funnell: And so how will that have an impact upon us?

Jonathan Borwein: One of the things is that mathematics is simultaneously what Gauss called 'the queen of science' and what Bertrand Russell called 'the most inaccessible of the arts'. We have a literature which has papers that were written 300 years ago that people still want to read. And assuming that they are even in English and finding everything out there, that's like hunting in a huge library. And then just the things that modern computers can do in terms of searching are fantastic.

But then more importantly in mathematics, nobody really wants, except for cultural reasons, to read 200-year-old medicine. At the other end, mathematics is many, many languages. So I can be a really good mathematician in the 5% of mathematics I understand, but it isn't going to help me speak to the people who speak different languages. So what we're discovering is that computer communication and collaborative techniques are allowing much more versatile teams, real teams of people with complimentary skills and they don't all have to be in the same physical location to try working with problems. And often it's having somebody from outside who doesn't come in with the same preconceptions who can make a breakthrough, but they can't make it until you have enough common language.

And then, just like in the rest of life, the computer is changing things. I don't know if you followed IBM Watson, the computer machine that beat the Jeopardy game champions?

Antony Funnell: Yes, we followed it on the program actually.

Jonathan Borwein: That's what I would have thought. And I have been (without much luck) talking to people at IBM as to whether they would let us try and build Watson for mathematics. I think that they are so busy doing it for medicine that we're not high on their horizon. But the ability in a subject like mathematics to work out how to answer questions when you don't even know what the question is...you know, often you need a graduate course in a subject to get into a literature in math, and the ability maybe 10 years from now for informal natural language help to be able to extract answers when you don't even know the right questions, that's very exciting.

Antony Funnell: And in one sense mathematicians have very much been leading the way in breaking down the silos, if you like, in embracing new technologies for collaboration, haven't they. Because I know a lot of other areas of science have struggled to get out of those silos.

Jonathan Borwein: I think you can see the glass as three-quarters full or three-quarters empty. So, for example, take a field like astrophysics. It's very well defined. Everyone knows who the leading astrophysicists are and where they work and who they are funded by. So, for example, for the last 20 years they've had really good archives, really good ways of interacting with themselves, but you're talking about maybe 300 or 400 people worldwide.

And you come to a field like mathematics or computing, well, how do you actually know how many of us there are and where we are and what we're doing. So in terms of...there's some projects like Polymath that people like Tim Gowers have been involved in. I think probably that is a radical departure for the sciences, and in that sense, again, mathematics is remarkable because you can't do world-class physics in 75 countries. There is no way you can have the equipment and have the environment to ask questions in many modern fields if you're not in the middle of an OECD country. But mathematics, particularly with the internet to help you now, you can be almost anywhere, and that of course changes both the challenge and the opportunities.

Awards ceremony MC: The Fields Medal was awarded to Terence Tao of the department of mathematics at the University of California at Los Angeles, for his contributions partial differential equations, combinatorics, harmonic analysis and additive number theory.

Antony Funnell: Terence Tao is our final guest today. He was a child prodigy and the recipient, as you've just heard, of the Fields Medal in 2006, the highest accolade in mathematics.

Now, we heard Gordon Clarke talking at the very beginning of the program about crowd-sourced online collaborations in maths, and Jon Borwein mentioned one such collaboration called the Polymath Project.

Well, Professor Tao has been one of the driving forces behind that project and indeed the whole notion of open collaborative maths.

Terence Tao: The Polymath Project started about five or six years ago by a friend of mine, Tim Gowers, and he just posted about asking what would a collaborative mathematics project look like, and it got a lot of responses. I think people were very excited. Mathematics has always been historically almost a secretive process where everyone is shut up in their offices and they are secretly working on these problems, either by themselves or with a few collaborators, and given that there are so many other open projects elsewhere, like Wikipedia and so forth, certainly had a great desire to try out something like this for mathematics.

So he started the first project four or five years ago, and he selected a problem which was quite difficult and it got a huge response, hundreds and hundreds of comments on the first few posts, it was quite chaotic in fact. So initially there were hundreds of people participating. Eventually once people understood what the problem was, there was a core of maybe six or seven people, mostly professional mathematicians, who ended up seriously working on the problem, and ended up solving it in about three months.

But the thing is that unlike most other collaborations in mathematics this was all done openly, so every time someone had a thought they would put it online in one of the blogs and then it would get discussed and batted around. And one of the most common feedback that we got after the project when it succeeded was that it was great that they'd solved the problem but perhaps one of the great values of this project was beginning mathematicians, like graduate students, could actually see for the first time the process of how people actually solve problems and how they actually get things done in mathematics, because all too often these things are presented as a fait accompli, as a solved problem. You only see the successes, you don't see the trial and error and the dead ends.

Antony Funnell: And is that still the case as far as you're concerned? Do you think the value of these sort of open collaborative projects where maths is concerned is still as much about the actual process as the end solution to a mathematical problem?

Terence Tao: Currently I think so, yes. We're still doing a few Polymath projects. I'm involved in one right now. But it's still only a very, very small fraction of al mathematical research that's going on nowadays. Most mathematics is still done in a very traditional way. So I don't think it's going to really revolutionise the common practice of research, at least not in the short term. But for revealing the process and also in getting younger mathematicians excited about certain types of problems and so forth, I think that is going to be the greater value, at least in the near term.

Antony Funnell: So taking what you've said there, and I hear what you're saying, is a collaborative approach, not necessarily just using the Polymath model, but is a more collaborative approach likely to occur in mathematics over time with demographic change?

Terence Tao: Yes, I believe so. There are other changes like this that have occurred very rapidly. For example, nowadays almost all mathematicians put their papers online on public servers for everyone else to read. They can get notified by email of when other papers in their field get released, and this is quite a change from, say, 20 years ago where people often so jealously guarded what they were working on and they wouldn't release a paper until it was accepted by a journal because they were afraid that maybe someone would steal their work or something. But there has been a big trend towards some openness, at least in recent time. I think part of that is generational, people are much more comfortable with the internet and sharing. So it could well be that there could be further cultures just like this.

Antony Funnell: And the shift to computer-based mathematics, what has that done for maths research?

Terence Tao: In some areas it has become extremely important. There are some parts of mathematics where now massive computer experiments are routinely used, both to figure out what is true and also how to solve these problems. In some areas there are some parts of mathematics that are very abstract and theoretical, they involve questions that even the fastest computers can't really touch. In my research I mostly use computers mostly as a communication tool and to learn about other people's research and so forth, but not so much in my own research, other than of course just to write my papers and things like that.

Antony Funnell: Why is that? Can I get you to be specific about why that is?

Terence Tao: Part of it I think is because to really use computers properly I'd have to learn how to program a lot better than I do right now, but I think also just the type of mathematics I do is also rather theoretical and abstract, and there are a lot of questions which you can't answer, even if you had really good computers at your disposal.

Antony Funnell: Is that going to change over time, do you think?

Terence Tao: There is a branch of mathematics called applied mathematics which is more focused on practical questions like can you get computers to speed up image search or voice recognition or whatever, and these people use computers very heavily, and this is an extremely fast growing area of mathematics and probably actually, to be honest, much more immediately useful than theoretical mathematics. So over time I think more and more fields of mathematics will have an applied labour to them.

Antony Funnell: Well, Professor Terry Tao from the department of mathematics at UCLA, thank you very much for joining us.

Credits

Comments (5)

mark :

16 Nov 2012 11:44:30pm

As an Enginner, now teaching High Scool maths for the last 3 years a lot of the ideas rang true. Kids just don't see the relevance of maths at high school. The applied maths content for harder maths in high schools is missing. Simple harmonic motion, projectile motion are considered applied topics but they are still centuries old?

Moderator :

25 Nov 2012 3:26:01pm

Hi Mark,

Thanks for a great comment. Really interesting to hear that many of the ideas in the show resonated with your experience of teaching maths. It will be interesting to see whether the maths curriculum changes over the next few years.

Alfred :

I know you said up the start of the program that the subject matter was difficult to present on radio, but I must say, I prefer programs that are reasonably specific.

What intrigues me is the use of computers for symbolic maths. It seems to me, from a point of symbolic manipulation, that computers basically become an extension of your memory storing the equations.

Maths though, is more than symbolic manipulation, although most maths books describe mathematical derivations and methods of rearranging equations. A big part of Mathematics, I think, is also about trying to model real world problems in mathematical form and understanding the equations that represent the behavior of objects in the real world.

What concerns me about using programs like Mathematica is that the understanding of how the mathematical algorithms work is hidden. I suppose it's a bit like trying to write music using a music notation package, you have to have an understanding of musical constructs as well, but I think there is something to be gained by understanding how symbolic manipulation can be represented and manipulated using computer algorithms.